Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak g=T_{e_G}G$ and $\mathfrak h=T_{e_H}H$. Let $\varphi:G\to H$ be a Lie [group homomorphism](/page/Group%20Homomorphism), and write
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\begin{align*}
d\varphi:=d\varphi_{e_G}:\mathfrak g\to\mathfrak h
\end{align*}
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for its differential at the identity. Then $d\varphi$ is a [Lie algebra](/page/Lie%20Algebra) homomorphism: for all $X,Y\in\mathfrak g$,